In his celebrated Erlangen Program in 1872, F. Klein opened a way to synthesize geometric objects based on group symmetry. Since then the notion of group has been playing signiﬁcant roles in the study of various geometries. Among them, fundamental is the so-called projective geometry, which is intimately related to that of vector spaces. Interrelations of geometric positions of ﬂat objects such as lines and planes in Euclidean spaces are described most aesthetically in the framework of projective geometry. The fundamental theorem of projective geometry then states that the three-point collinearity is enough to recover the linear group structure behind them. Its importance is not just restricted within purely mathematical subjects and we shall review here, in quantum theory and special relativity, two fundamentals in physics, how their symmetries can be realized as linear groups as applications of the fundamental theorem.
Symmetry is such a vast subject that its overall description requires lots of words. As a perspective course, I present here an interdisciplinary topic which ranges from geometry to algebra with physical backgrounds. It is just a tiny part of the subject but still provides good impetus to the typical way of thinking of symmetry.